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On regular variation for infinitely divisible random vectors and additive processes

Part of: Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Henrik Hult  and
Filip Lindskog
Henrik Hult*
Affiliation:
Cornell University
Filip Lindskog*
Affiliation:
KTH Stockholm
*
∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, 414A Rhodes Hall, Ithaca, NY 14853, USA. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, KTH, 100 44 Stockholm, Sweden.
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Abstract

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We study the tail behavior of regularly varying infinitely divisible random vectors and additive processes, i.e. stochastic processes with independent but not necessarily stationary increments. We show that the distribution of an infinitely divisible random vector is tail equivalent to its Lévy measure and we study the asymptotic decay of the probability for an additive process to hit sets far away from the origin. The results are extensions of known univariate results to the multivariate setting; we exemplify some of the difficulties that arise in the multivariate case.

Keywords

Multivariate regular variationinfinitely divisible distributionadditive processLevy process

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60G70: Extreme value theory; extremal processes 60E07: Infinitely divisible distributions; stable distributions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 134 - 148
Copyright
Copyright © Applied Probability Trust 2006 

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